Compliant actuators are more advantageous than stiff actuators in some circumstances, for example, unstructured environment robots and rehabilitation robots. Compliant actuators are more adaptive and safe. Constant stiffness compliant actuators have some limitations in impedance and bandwidth. Variable stiffness actuators improve their performance owing to introducing an extra motor to tune the stiffness of the actuators. However, they also have some limitations such as the bulky structure and heavy weight. It was also found that there are some waste functions existing in the current variable stiffness actuators and that the fully decoupled position control and stiffness tune are not necessary, because there exist some regular phenomena during most circumstances of human interaction with the robots which are “low load, low stiffness and high load, high stiffness”. In this paper, a design method for nonlinear stiffness compliant actuator was proposed which performed the predefined deflection-torque trajectory of the regular phenomenon. A roller and a cantilever which has special curve profile constitute the basic mechanical structure of the nonlinear stiffness compliant actuators. An error compensation method was also proposed to analyze the stiffness of elastic structure. The simulation results proved that the proposed method was effective in designing a predefined nonlinear stiffness compliant actuator.
Traditional actuators used in robotics or machines are stiff actuators which could obtain good position control and bandwidth. However, since gear boxes are usually used in stiff actuators, the backdrivability is always poor, which is not suitable in some circumstances, for example, human-machine interaction, bioinspired robots and unstructured environment robots, and rehabilitation robots. The compliant actuator using an elastic component between the actuator and load was first proposed by Pratt in MIT [
From the early SEA with a linear spring to the later one with a torsional spring [ Change transmission between the load and elastic structures. AwAS [ Change the preload of the spring. FSJ [ Change the physical structure of springs. Jack Spring [
In general, most of variable stiffness actuators use two actuator units: one for position control and the other for the stiffness regulation, which makes the position control and stiffness regulation independent. Of course, variable stiffness actuator is flexible to adjust the stiffness whatever the position of rotor is. However, it also leads to the bulky and complicated mechanical structure [
Few researchers studied nonlinear stiffness SEA without an additional motor. Migliore et al. developed a nonlinear stiffness SEA with antagonistic nonlinear elastic springs [
This functionality is useful for human to obtain dexterous performance with low load and stability with high load [
In this paper, a new design method of compliant actuators with predefined nonlinear stiffness trajectory of “low load, low stiffness and high load, high stiffness” was proposed. The basic structure of the proposed mechanism consisted of a roller, an elastic component, and a contact part. One joint of 3-DOF (degrees of freedom) rehabilitation robot for shoulder training was introduced as the compliant actuator with defined nonlinear stiffness. Theory deduction and resolution process using software were described in detail. The simulation and experiments were also conducted.
In this part, one compliant joint of a rehabilitation robot for shoulder complex was introduced which used the proposed nonlinear elastic elements. This robot was designed with 3 DOF (RRP configuration) by International Center of Advanced Mechanisms and Robotics (iCAMAR) of Tianjin University. All of these joints installed the nonlinear elastic elements and the mechanical design of the first joint was introduced as an example which is shown in Figure
CAD model of the first joint of rehabilitation robot.
The key issue of the design of a nonlinear stiffness compliant actuator is to design elastic mechanisms with nonlinear stiffness. In this paper, a very basic and common elastic structure was adopted as the elastic element, namely, cantilever structure, which is easy to design and accurate to calculate. The main idea of topology design is to propose an elastic element composed of the elastic part and the contact part which has a special curve profile contacting with a roller. The impedance in the direction of roller motion could be changed when the slope of the tangent plane between the roller and the contact part varied, and then the mechanical stiffness of system changed.
The core of designing the nonlinear stiffness compliant actuator is the nonlinear elastic mechanism. As we investigated from the current literature, there is no method to obtain the mechanism whose stiffness matches the predefined nonlinear stiffness trajectory. In this paper, we proposed a simple mechanism to achieve it, which consists of an elastic element and a roller. The elastic element consists of two parts: the elastic part and the contact part. The elastic part is a uniform cantilever which will generate deflection and rotation when it is given a certain external force. The contact part has a specific curve profile.
Figure
The schematic diagram of force analysis of the proposed mechanism.
The force analysis in the mechanism is also shown in Figure
To simplify analysis of the deflection of the elastic element, the elastic element was analyzed by divided it into two parts. One is a simple cantilever and the other is variable cross section cantilever. Generally speaking, the deflection of elastic element mainly happened on the elastic part and the deflection of the contact part is very little. So firstly analyze the deflection of elastic element by assuming that the contact part is rigid body. Then calculate the curve profile of contact part on the elastic element and conduct the simulation via ANSYS software. Finally, compare the desired torque-deflection trajectory with simulation results, and a method of error compensation was proposed to decrease the error of obtained torque-deflection caused by the assumption of rigidity of the contact part.
In this research, the elastic element is required to deform within a small deflection, so that the mechanical system can be of good elastic performance. Therefore, the deflection of the elastic element can be described using differential equations of the deflection curve. In Figure
According to the differential equations of the deflection curve, the deflection angle and deflection of the extreme of elastic component can be described as the following equations:
The geometric relationships shown in Figure
Equations (
The slope of contact point B before the deformation of the elastic element can be expressed by (
The length of the semimajor axis and the semiminor axis of ellipse can be described by (
Assume that an external load applied to the system is
For a predefined deflection-torque relationship of “low load, low stiffness and high load, high stiffness” which cannot be described using a mathematical function, a numerical method was proposed to solve the adequate curve profile by using MATLAB software. The following part shows the procedure of process for solution of the following available point on the curve profile of the contact part: Initialization For Obtain rotational angel and torque Resolve equations using Get next contact point ( End
An interval (iteration step) which affects the accuracy of curve profile was chosen firstly to obtain the number of steps. According to (
Since the combination of a cantilever and contact part with a curve profile was used as elastic element of nonlinear elastic system, it is inevitable to refer to the elastic deflection calculation of variable cross section. However, it is difficult to describe this deflection precisely with some algebraic equations.
A simple resolution was proposed to describe the relationship of the deflection and torque in this case. Firstly, analyze the deflection and deflection angle of the cantilever while assuming the contact part is rigid body so that the preliminary shape of the curve profile of the contact part could be derived as mentioned above. According to the simulation results, the derived trajectory of torque-deflection is always a little larger than that derived via theory calculation where the same torque was set. It is not difficult to analyze the reason that the contact part also deflects tinily in horizontal and vertical directions. So some errors should be generated via calculating the geometric equations. Meanwhile, the coordinates of the curve profile of undeformed elastic element were derived via those of deformed elastic element, which makes the derived torque-deflection trajectory of simulation always lower than that derived from the theoretical calculation.
Errors analysis of theoretical calculation and simulation results is shown in Figure
Errors analysis of theoretical calculation and simulation results.
The stiffness of compliant actuators can be obtained by the curve profile of contact part. However, the inherent stiffness of elastic part has an effect on the range of the stiffness of compliant actuator. It is essential to determine the inherent stiffness firstly. When the inherent stiffness of elastic part is high, the range of the compliant actuator is large. However, the high stiffness generally induces large stress in the elastic element owing to the fact that the inherent stiffness is not fully involved. The inherent stiffness of the elastic part depends on its dimension, and the shape is necessary to be optimized to acquire the appropriate stiffness.
Take initial inherent stiffness of the extreme of the elastic part as optimal object. Its value is higher than the required value which can be calculated via the desired torque-deflection trajectory. The design and optimization of details of contact part were implemented. When the derived shape of the elastic part is available in terms of allowable stress and assembly conditions, the inherent stiffness would be determined. Otherwise, the next inherent stiffness and optimization process would be repeated for the available shape of the elastic part.
Three groups of elastic elements are installed on a plane and each bears the maximum forces 12 Nm within 1.9 degrees in our proposed compliant robotic joint. The dimensions of the cantilever are design variables and the stress is the status variable. The chosen material is 50 CrVA (elasticity modulus 2.06 Gpa, poisson ratio: 0.29, and yield strength: 1.32 Gpa).
To obtain the optimized structure of the elastic element, the genetic algorithm was used under the constraint conditions of design variables and status variables by using the commercial software ANSYS. The assembly constraint conditions in our proposed structure are 23 mm < length < 25 mm, 3 mm < width < 3.5 mm and 3 mm < height < 5 mm, and stress < 900 Mpa. The obtained stiffness of the extreme of the elastic part is 1400 N/mm. Table
The result of optimization.
Design variable | Optimized value (mm) |
---|---|
| 24.4 |
| 3.4 |
| 4.9 |
The thickness of the contact part is related to the simulation results of nonlinear stiffness trajectory. Figure
The thickness of the contact part in the elastic element.
The torque-deflection trajectory with different thicknesses.
For the circumstance of high precision requirement to stiffness characteristic, the error compensation has to be considered, which is induced by the assumption of rigid body of contact part. The method of error compensation is mentioned in Section
Uncompensated torque-deflection trajectory.
The compensated torque-deflection trajectory with the same desired trajectory as Figure
Compensated torque-deflection trajectory.
Figure
Relative errors using and not using compensation method.
In this paper, a new design methodology of compliant mechanisms whose stiffness matches the predefined deflection-torque trajectory of “low load, low stiffness and high load and high stiffness” and an error compensation method were proposed. The assumption that the contact part of elastic element was viewed as rigid body is almost effective in getting the predefined nonlinear stiffness trajectory mentioned above. For obtaining more accurate trajectory, an error compensation method was proposed. According to the simulation results, the proposed method is highly effective in decreasing the relative errors. One joint of the rehabilitation robot for shoulder complex developed by ourselves was illustrated. This research achievement will be useful to design new compliant actuators used in bioinspired robotics, rehabilitation robotics, and so on.
The length of cantilever
The radius of roller
The radius of circular trajectory of the roller movement
The length of the semimajor axis of ellipse C
The length of the semiminor axis of ellipse C
The width of cross section of the cantilever
The height of cross section of the cantilever
The angle between the slope plane at the contact point and
The rotational angle of roller motion in terms of the rotatory component where the elastic element was installed
Deflection of the end of the elastic part
Deflection angle of the end of the elastic part
The coordinates of point A before the deformation of the elastic element
The coordinates of point B before the deformation of the elastic element
The coordinates of point B after the deformation of the elastic element
The current coordinates of the roller center
The initial coordinates of the roller center
The component force of contact forces between the roller and the elastic element
The external torque.
The authors declare that they have no competing interests.
The financial support from the Natural Science Foundation of China (Project no. 51475322) and the Programme of Introducing Talents of Discipline to Universities (“111 Program”) under Grant no. B16034 and Open Fund of Key Laboratory of Mechanism Theory and Equipment Design of Ministry of Education of China is greatly acknowledged.